Since the Hawking-Page phase transition [1]} of an AdS spacetime was proposed by Hawking and Page in 1983, which was explained to a confinement/deconfinement phase transition in gauge theory in Ref. [2]} and could also be understood as a solid/liquid phase transition [3]} by regarding the cosmological constant as pressure \(P=-\frac{\Lambda }{8\pi }=\frac{(n-1)(n-2)}{16\pi l^2}\) , black hole thermodynamics has attracted lots of attention. Subsequently the phase transition in the extended AdS/dS phase space has been widely considered [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [7]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}. Based on the similarity of thermodynamic behaviors for black holes and ordinary systems, the microstructure of black holes system was investigated by the Ruppeiner scalar curvature [21]} following the Ruppeiner geometry [34]}. The number density of the speculative black hole molecules was introduced to examine its phase transition and microstructure. This method was quickly generalized to other black holes [35]}, [36]}. Furthermore from the view point of the Gibbs free energy landscape, the dynamic properties of various black hole phase transitions were studied in Refs. [37]}, [38]}, [39]}, [40]}, [41]}. Since the phase transition of black holes is more important in black hole thermodynamics, there exists a natural question how to probe the information of the black hole phase transition.
